Calculate the half-life of uranium if its initial mass of 1 g is reduced by 2 mg in 5 years by disintegration into thorium and an `alpha` -particle.

Explanation

Given, initial mass, `m_(0)=1` g
Mass reduced in disintegration `=m_(0)-m=2 mg = 0.002 g`
`therefore m=m_(0) - 0.002 = 1- 0.002 = 0.998`g
where m is the mass of sample after disintegration
Also, `m = m_(0)e^(-lambdat)`
where `lambda` is the decay constant
and t is the time taken for disintegration = 5 years
`rArr e^(lambdat) =(m_(0))/m`
`lambda t = log_(e)(m_(0)/m)`
`lambda =2.303/t log_(10)m_(0)/m`
`=2.303/5 log_(10)1/0.998`
`=4.145 xx 10^(-4) "year"^(-1)`
Half-life is given by
`T_(1//2) = 0.693/lambda = 0.693/(4.1454 xx 10^(-4)) = 1,672` years.